Find the difference quotient of $LaTeX: \displaystyle f(x)=- 7 x^{3} - 9 x^{2} - 7 x - 1$ .

The difference quotient is $LaTeX: \displaystyle \frac{f(x+h)-f(x)}{h}$ . Evaluating $LaTeX: \displaystyle f(x+h)=- 7 h - 7 x - 7 \left(h + x\right)^{3} - 9 \left(h + x\right)^{2} - 1$ and expanding gives $LaTeX: \displaystyle f(x+h)=- 7 h^{3} - 21 h^{2} x - 9 h^{2} - 21 h x^{2} - 18 h x - 7 h - 7 x^{3} - 9 x^{2} - 7 x - 1$ Evaluating the difference quotient gives $LaTeX: \displaystyle \frac{(- 7 h^{3} - 21 h^{2} x - 9 h^{2} - 21 h x^{2} - 18 h x - 7 h - 7 x^{3} - 9 x^{2} - 7 x - 1)-(- 7 x^{3} - 9 x^{2} - 7 x - 1)}{h}$ Simplifying gives $LaTeX: \displaystyle \frac{- 7 h^{3} - 21 h^{2} x - 9 h^{2} - 21 h x^{2} - 18 h x - 7 h}{h}=- 7 h^{2} - 21 h x - 9 h - 21 x^{2} - 18 x - 7$